CERN-TH/97-221

hep-th/9709174

Finite Energy Solutions in Three-Dimensional

Heterotic String Theory

Michele Bourdeau and Gabriel Lopes Cardoso 1

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

ABSTRACT

We show that a large class of supersymmetric solutions to the low-energy

effective field theory of heterotic string theory compactified on a seven torus

can have finite energy, which we compute. The mechanism by which these

solutions are turned into finite energy solutions is similar to the one occurring

in the context of four-dimensional stringy cosmic string solutions. We also

describe the solutions in terms of intersecting eleven-dimensional M-branes,

M-waves and M-monopoles.

CERN-TH/97-221

September 1997

1Email: bourdeau@mail.cern.ch, cardoso@mail.cern.ch

1 Introduction

A large class of supersymmetric soliton solutions in string theory have by now been

constructed in various dimensions (for a review see for instance [1, 2, 3] and references

therein), as these play a fundamental role in duality studies. While most of the recent

work on supersymmetric solutions in string theory has been done in dimensions higher or

equal to four, some heterotic supersymmetric solutions have now been determined in three

space–time dimensions [4, 5]. The solutions presented in [4, 5] are static supersymmetric

solutions of the low-energy effective field theory of heterotic string theory compactified

on a seven-torus, which is described by a three dimensional supergravity theory with

eight local supersymmetries [6, 4].

A particular class of such heterotic solutions in three dimensions can be obtained [4] by

compactifying the four-dimensional string solutions of [7, 8] on a circle. In [4] it was shown

that the resulting three dimensional solutions can be turned into finite energy solutions

by utilizing a mechanism first discussed in the context of four-dimensional stringy cosmic

string solutions [9]. It was further conjectured in [4] that this mechanism should also

apply to other three-dimensional supersymmetric solutions. We will see that the same

mechanism can indeed be used for turning the solutions constructed in [5] into finite

energy solutions.

The construction of the supersymmetric solutions given in [5] was achieved by solving

the associated Killing spinor equations in three dimensions. These Killing spinors have

a priori 16 real degrees of freedom which, however, get reduced by imposing certain

contraints specific to each of the solutions. Up to three such independent conditions

(m = 1, 2, 3) can be imposed on the Killing spinors, resulting in Killing spinors with 1/2m

of 16 real degrees of freedom. The associated solutions were referred to as preserving 1/2m

of N = 8, D = 3 supersymmetry.

The solutions constructed in [5] are, however, only valid asymptotically, that is at large

spatial distances. Thus, they should get modified in such a way as to render them well

behaved at finite distances [4]. We will see that this is indeed possible by turning them

into finite energy solutions.

The solutions constructed in [4, 5] have a ten-dimensional heterotic interpretation in

terms of intersections of fundamental strings, NS 5-branes, waves and Kaluza–Klein

monopoles (or, equivalently, an eleven dimensional interpretation in terms of intersections

of M-branes, M-waves and M-monopoles). Let us for instance consider the solutions

constructed in [5]. They fall into two classes. Namely, they either carry one or two electric

1

charges. We will show below that the solutions carrying one electric charge have a ten-

dimensional interpretation in terms of a wave and up to three orthogonally intersecting

NS 5-branes. The solutions carrying two electric charges, on the other hand, have a

ten-dimensional interpretation in terms of a fundamental string, a wave and up to three

orthogonally intersecting NS 5-branes as well as up to three Kaluza–Klein monopoles.

On the other hand, the three-dimensional solutions obtained [4] by compactifying the

four-dimensional heterotic string solutions of [7, 8] on a circle have a ten-dimensional

interpretation in terms of a fundamental string and up to three orthogonally intersecting

NS 5-branes. We note here that the solutions constructed in [4, 5] should also have an

equivalent description in terms of configurations of type IIB p-branes (such as the 7-brane

of [10]) wrapped around K3× T3.

The ten dimensional space–time line element describing orthogonally intersecting strings,

5-branes, waves and Kaluza–Klein monopoles are given in terms of harmonic functions

which depend on some of the overall transverse directions. These line elements, when

compactified down to three dimensions, give rise to three dimensional space–time line

elements which are again given in terms of the same harmonic functions. In three space–

time dimensions, a harmonic function H(z, z) does not however get determined by the

condition of it being harmonic. Denoting the two spatial dimensions by z and z, one has

∂z∂zH = 0→ H = f(z) + f(z) , (1.1)

where f(z) is an a priori arbitrary holomorphic function. Thus, we expect that those het-

erotic supersymmetric three-dimensional solutions which have an M-theory description in

terms of orthogonally intersecting M-branes, M-monopoles and M-waves should also be

expressed in terms of holomorphic functions f(z). Demanding that these supersymmetric

solutions have a certain behaviour at spatial infinity (z → ∞) determines the form of

f(z) at large z. For instance, the solutions presented in [5] have an asymptotic behaviour

corresponding to f(z) ∝ ln z. At finite distance these asymptotic solutions become ill-

defined and so need to be modified. The associated corrections will all be encoded in f(z).

Requiring the solutions to have finite energy as well as the above asymptotic behaviour

will determine f(z) to be given by f(z) ∝ j−1(z). The resulting modified solutions are

then well-behaved at finite distances. Asymptotically, the associated coupling constant

g2 = e2φ is weak, whereas at finite distances it becomes strong.

This paper is organised as follows. In section 2 we review some properties of the low-

energy effective action of heterotic string theory compactified on a seven-dimensional

torus [4]. In section 3 we give the Killing spinor equations associated to the three-

dimensional heterotic low-energy effective Lagrangian and we present some results.

2

In section 4 we first review some of the static supersymmetric solutions carrying two

electric charges, whose asymptotic form was given in [5]. These solutions are labelled

by an integer n with n = 1, 2, 3, 4. We then show how these solutions can be rewritten

in terms of holomorphic functions f(z), and check that the associated three-dimensional

Killing spinor equations are solved for any arbitrary holomorphic f(z). Demanding that

these solutions have finite energy as well as the asymptotic behaviour that was found in

[5] determines f(z) = n+1πf(z) to be given by j(f(z)) = z. The dilaton is, for any n,

determined to be

e−φ = f(z) + f(z) . (1.2)

Next, we compute the energy E carried by these solutions and find that (in units where

8πGN = 1) E = 2n π6. The solutions presented in [5] describe one-center solutions. They

can be straightforwardly generalized to multi-center solutions via j(f(z)) = P (z)/Q(z)

[9], where P (z) and Q(z) are polynomials in z with no common factors.

In section 5, we repeat the analysis given in section 4 for some of the static solutions now

carrying one electric charge with asymptotic form given in [5]. These solutions are again

labelled by an integer n with n = 1, 2, 3, 4. As in the case of two electric charges, we

show how these solutions can be rewritten in terms of holomorphic functions f(z), and

we check again that the associated Killing spinor equations are solved for any arbitrary

holomorphic f(z). As before, demanding that these solutions have finite energy as well

as the asymptotic behaviour found in [5] determines f(z) = n+22πf(z) to be again given

by j(f(z)) = z. This time, however, the dilaton is, for any n, determined to be

e−2φ = f(z) + f(z) . (1.3)

We find that the energy E associated with these solutions is (in units where 8πGN = 1)

E = n π6

. This is half of the amount carried by the solutions with two electric charges.

These solutions can again be straightforwardly generalized to multi-center solutions via

j(f(z)) = P (z)/Q(z).

In section 6, we present the eleven-dimensional interpretation in terms of orthogonally

intersecting M-branes, M-waves and M-monopoles for the solutions discussed in sections

4 and 5.

Finally, in section 7, we present our conclusions.

We use the same conventions as in [5].

3

2 The three-dimensional effective action

The effective low-energy field theory of the ten-dimensional heterotic string compacti-

fied on a seven-dimensional torus is obtained from reducing the ten-dimensional N = 1

supergravity theory coupled to U(1)16 super Yang–Mills multiplets (at a generic point

in the moduli space) [11, 12, 4]. The massless ten-dimensional bosonic fields are the

metric G(10)MN , the antisymmetric tensor field B

(10)MN , the U(1) gauge fields A

(10)IM and the

scalar dilaton Φ(10) with (0 ≤ M,N ≤ 9, 1 ≤ I ≤ 16). The field strengths are

F(10)IMN = ∂MA

(10)IN − ∂NA

(10)IM and H

(10)MNP = (∂MB

(10)NP −

12A

(10)IM F

(10)INP )+ cyclic permuta-

tions of M,N, P .

The bosonic part of the ten-dimensional action is

S ∝∫d10x

√−G(10)e−Φ(10)

[R(10) +G(10)MN∂MΦ(10)∂NΦ(10)

−1

12H

(10)MNPH

(10)MNP −1

4F

(10)IMN F (10)IMN ]. (2.1)

The reduction to three dimensions [6, 12, 4] introduces the graviton gµν , the dilaton

φ ≡ Φ(10) − ln√

detGmn , with Gmn the internal 7D metric, 30 U(1) gauge fields A(a)µ ≡

(A(1)mµ , A(2)

µm, A(3)Iµ ) (a = 1, . . . , 30, m = 1, . . . , 7, I = 1, . . . , 16) , where A(1)m

µ are

the 7 Kaluza–Klein gauge fields coming from the reduction of G(10)MN , A

(2)µm ≡ Bµm +

BmnA(1)nµ + 1

2aImA

(3)Iµ are the 7 gauge fields coming from the reduction of B

(10)MN and

A(3)Iµ ≡ AIµ − a

ImA

(1)mµ are the 16 gauge fields from A

(10)IM .

The field strengths F (a)µν are given by F (a)

µν = ∂µA(a)ν − ∂νA

(a)µ . Finally, B

(10)MN induces

the two-form field Bµν with field strength Hµνρ = ∂µBνρ −12A(a)µ LabF

(b)νρ + cyclic

permutations.

The 161 scalars Gmn, aIm and Bmn can be arranged into a 30×30 matrix M (we use here

the conventions of [12])

M =

G−1 −G−1C −G−1aT

−CTG−1 G + CTG−1C + aTa CTG−1aT + aT

−aG−1 aG−1C + a I16 + aG−1aT

, (2.2)

where G ≡ [Gmn], C ≡ [12aIma

In +Bmn] and a ≡ [aIm].

We have MLMT = L, MT = M, L−1 = L, where

L =

0 I7 0

I7 0 0

0 0 I16

. (2.3)

4

We use the following ansatz for the Kaluza–Klein 10D vielbein EAM and inverse vielbein

EMA , in the string frame

EAM =

eφeαµ A(1)mµ eam

0 eam

, EMA =

e−φeµα −e−φeµαA

(1)mµ

0 ema

, (2.4)

where eam is the internal and eαµ the space–time vielbein in the Einstein frame (the relation

between string metric Gµν and Einstein metric gµν in three dimensions is Gµν = e2φgµν).

The three-dimensional action in the Einstein frame is then [12, 4],

S =1

4

∫d3x√−gR− gµν∂µφ∂νφ−

1

12e−4φgµµ

′gνν

′gρρ

′HµνρHµ′ν′ρ′

−1

4e−2φgµµ

′gνν

′F (a)µν (LML)abF

(b)µ′ν′ +

1

8gµνTr (∂µML∂νML) , (2.5)

where a = 1, . . . , 30.

This action is invariant under the O(7, 23) transformations

M → ΩMΩT , A(a)µ → ΩabA

(b)µ , gµν → gµν , Bµν → Bµν , φ→ φ, ΩTLΩ = L,

(2.6)

where Ω is a 30× 30 O(7, 23) matrix.

The equations of motion for A(a)µ , φ, Hµνρ and gµν are, respectively,

∂µ(e−2φ√−g(LML)abF(b)µν) +

1

2e−4φ√−g Lab F

(b)µρ H

νµρ = 0 , (2.7)

DµDµφ+

1

4e−2φF (a)

µν (LML)abFµν(b) +

1

6e−4φHµνρHµνρ = 0 , (2.8)

∂µ(√−ge−4φHµνρ) = 0 , (2.9)

Rµν = ∂µφ∂νφ+1

2e−2φF (a)

µρ (LML)abFρ(b)ν −

1

8Tr (∂µML∂νML) (2.10)

−1

4e−2φgµνF

(a)ρτ (LML)abF

ρτ(b) +1

4e−4φHτσ

µ Hντσ −1

6gµνe

−4φHτσρHτσρ .

We will, in the following, consider backgrounds where Hµνρ = 0.

Consider a static solution with space–time line element of the form

ds2 = −dt2 + gxx (dx2 + dy2) = −dt2 + 2gzz dzdz , (2.11)

where z = x+ iy. Its energy E can be computed from (2.7)–(2.10) and is given by

E =1

8πGN

∫d2x gxxTtt =

i

8πGN

∫dzdz gzzTtt =

i

16πGN

∫dzdz gzzR , (2.12)

where 8πGN Tµν = Rµν −12gµνR. Here GN denotes the three-dimensional gravitational

constant. We will in the following set 8πGN = 1.

5

From the equations of motion for the gauge fields A(a)µ (2.7) one can define a set of scalar

fields Ψa, a = 1, . . . , 30, through [4]

√−ge−2φgµµ

′gνν

′(ML)abF

(b)µ′ν′ = εµνρ∂ρΨ

a,

F (a)µν =1√−g

e2φ(ML)abεµνρ∂ρΨ

b . (2.13)

Then, from the Bianchi identity εµνρ∂µF(a)νρ = 0,

Dµ(e2φ(ML)ab∂µΨb) = 0. (2.14)

Following [4], the charge quantum numbers of elementary string excitations are charac-

terized by a 30-dimensional vector ~α ∈ Λ30. The asymptotic value of the field strength

F (a)µν associated with such an elementary particle can be calculated to be [4]

√−gF (a)tr ' −

1

2πe2φMabα

b. (2.15)

The asymptotic form of Ψa is then

Ψa ' −θ

2πLabα

b + constant. (2.16)

It can be shown [4] that the matrix M , the Ψ’s and the dilaton can be assembled into

a matrix M describing the coset O(8,24)O(8)×O(24)

. The low energy effective three-dimensional

field theory is then actually invariant under O(8, 24) transformations. An O(8, 24;Z)

subgroup of this group is a symmetry of the full string theory [4].

3 The Killing spinor equations

In ten dimensions, the supersymmetry transformation rules for the gaugini χI , dilatino

λ and gravitino ψM are, in the string frame, given by [13, 14, 15, 16, 17]

δχI =1

2F IMNΓMNε ,

δλ = −1

2ΓM∂MΦε +

1

12HMNPΓMNPε ,

δψM = ∂Mε+1

4(ωMAB −

1

2HMAB)ΓABε . (3.1)

These equations were reduced to three dimensions in the Einstein frame in [5]. In the

following, we restrict ourselves to backgrounds withHµνρ = 0 and aIm = 0. The associated

three-dimensional Killing spinor equations become

δχI =1

2e−2φF (3)I

µν γµνε,

6

δλ = −1

2e−φ∂µφ+ ln det eamγ

µ ⊗ I8 ε+1

4e−2φ[−BmnF

(1)nµν + F (2)

µνm]γµνγ4 ⊗ Σmε

+1

4e−φ∂µBmnγ

µ ⊗ Σmnε ,

δψµ = ∂µε+1

4ωµαβγ

αβε+1

4(eµαe

νβ−eµβe

να)∂νφγ

αβε+1

8(ena∂µenb−e

nb ∂µena)I4 ⊗ Σabε

−1

4e−φ[ema F

(2)µν(m) − emaF

(1)mµν ]γνγ4 ⊗ Σaε−

1

8∂µBmn I4 ⊗ Σmnε

+1

4e−φBmnF

(1)nµν γνγ4 ⊗ Σmε ,

δψd = −1

4e−φ(emd ∂µema+ema ∂µemd)γ

µγ4 ⊗ Σaε+1

8e−2φemd BmnF

(1)nµν γµνε

+1

4e−φemd e

na∂µBmnγ

µγ4 ⊗ Σaε−1

8e−2φ[ emdF

(1)mµν + emd F

(2)µνm ]γµνε , (3.2)

where δψd ≡ emd δψm denotes the variation of the internal gravitini.

Static solutions to the Killing spinor equations can be constructed by setting the su-

persymmetry variations of the fermionic fields to zero. This ensures that the bosonic

configurations so obtained are supersymmetric.

The supersymmetric static solutions we will be discussing in the following sections will

either carry one or two electric charges. They have the following space–time line elements:

ds2 = −dt2 +Hcn(ω, ω)dωdω , c = 1, 2 , n = 1, 2, 3, 4 , (3.3)

where ω = r + iθ, and where the solutions carrying one (two) electric charge have c = 1

(c = 2). H(ω, ω) denotes a harmonic function:

∂ω∂ωH = 0 −→ H(ω, ω) = f(ω) + f(ω) . (3.4)

The dilaton is found to be

e−2φ = Hc . (3.5)

In all cases, we will make the following ansatz for the Killing spinors

ε = ε⊗ χ , (3.6)

where εT = (ε1, ε2, ε3, ε4) is a SO(1, 2) spinor and χ is a SO(7) spinor of the internal

space. We will be able to solve the Killing spinor equations by imposing the following

two conditions on ε [5]:

γ1ε = ip γ2 ε , γ1ε = p Jγ2γ4 ε , (3.7)

7

where p = ± , p = ± . Here we used that γµ = γαeµα , α = 0, 1, 2.

It follows that

ε = ε(ω, ω)

ip

1

p

−ipp

, (3.8)

and, hence, ε contains only two real independent degrees of freedom. χ, on the other

hand, contains eight real degrees of freedom; thus there are a priori a total of 16 real

degrees of freedom. These will be further reduced by conditions on χ specific to each

case considered. Up to three such independent conditions (m = 1, 2, 3) can be imposed

on χ, thereby allowing for the construction of solutions whose Killing spinors have 1/2m

of 16 real degrees of freedom. The solutions with n = 1 and n = 2 have Killing spinors

with 8 and 4 real degrees of freedom, respectively. The solutions with n = 3 and n = 4

both have Killing spinors with 2 real degrees of freedom.

In all cases considered in the following, we find that

ε = eφ2 = H−

c4 , (3.9)

up to a multiplicative constant.

4 Supersymmetric solutions carrying two electric charges

Here, we will consider the class of solutions carrying two electric charges presented in

[5]. The solutions given there are well behaved only at large spatial distances, and hence

need to be modified at finite distances.

The solutions in this class are labelled by an integer n (n = 1, 2, 3, 4). The associated

dilaton was given by (in a specific coordinate system) [5]

eφ =n+ 1

a ln r, (4.1)

whereas the associated space–time line element was

ds2 = −dt2 +a2

r2

(a ln r

n+ 1

)2n

(dr2 + r2dθ2). (4.2)

Here, a = n+12π

√|αiαi+7|, where αi and αi+7 denote the two electric charges. The i-

th component of the internal metric Gmn was given by Gii = | αiαi+7|. The gauge fields

8

strengths, or equivalently Ψi and Ψi+7, were taken to have the asymptotic form given in

(2.16). Introducing complex coordinates, ω = a ln z = a(ln r + iθ), then yields

ds2 = −dt2 +H2ndωdω , e−φ = H , (4.3)

where H = a ln rn+1

= ω+ω2(n+1)

. Inspection of (4.1) shows that the solution becomes ill defined

as r → 1 and thus it should get modified at finite distances. The asymptotic form of H

suggests the replacement

H =ω + ω

2(n+ 1)−→ H = f(ω) + f(ω) . (4.4)

Similarly, the asymptotic form of the Ψi given in (2.16) suggests the replacement

Ψi =αi+7

4πai(ω − ω) −→ Ψi =

(n+ 1)αi+7

2πai(f − f) ,

Ψi+7 =αi

4πai(ω − ω) −→ Ψi+7 =

(n+ 1)αi2πa

i(f − f) . (4.5)

Denoting the imaginary part of f by Ψ, Ψ = −i(f − f), we have

f =1

2

(e−φ + iΨ

). (4.6)

Using Gii = | αiαi+7|, we can rewrite Ψi and Ψi+7 in the following way

Ψi = −ηαi+7

√GiiΨ , Ψi+7 = −ηαi

√GiiΨ , (4.7)

where ηαi+7 = −ηαi denotes the sign of the two charges αi and αi+7. Thus we see that

the Ψ’s can be reexpressed in terms of the internal metric and of Ψ.

Using that ∂ωf = 0 it can be shown that the associated field strengths are given by

F(1)itβ = ηαi

√Gii

∂βH

H2, F

(2)tβ i = −ηαi

√Gii

∂βH

H2, β = r, θ , (4.8)

where ω = a(ln r + iθ) = r + iθ.

We now discuss the solutions in more detail. Let us first consider the case where n = 1.

The associated Killing spinor has 1/2 of 16 real degrees of freedom. The solution is

characterised by the fact that the internal vielbein eam is diagonal and constant and that

Bmn = 0 [5]. The solution presented in [5], which has line element (4.2), should be

modified so as to render the solution well behaved at finite distances. This modification

is given by (4.3), (4.4) and (4.5). It can then be checked that the Killing spinor equations

(3.2) are satisfied for any arbitrary holomorphic f . The Killing spinor is given by (3.6)

and (3.8) subject to one additional condition on χ given in [5]. Thus, the requirement of

supersymmetry alone does not determine f .

9

Next, consider the case where n = 2. The associated Killing spinor has 1/4 of 16 real

degrees of freedom. The solution is characterised by the fact that now there is one non-

constant off-diagonal entry in the internal metric as well as one non-constant entry in the

Bmn-matrix. As an example, consider the case where the two electric charges are taken

to be α4 and α11, and where the background fields Gmn and Bmn are given by [5]

G11 G12 0 0 · · · 0G21 G22 0 0 · · · 00 0 G33 0 · · ·

0 0 0 G44 0...

.... . .

0 · · · 0 G77

=

g2 −g2Υ2 0 0 · · · 0−g2Υ2 |

α9

α2|+ g2Υ2

2 0 0 · · · 00 0 G33 0 · · ·

0 0 0 G44 0...

.... . .

0 · · · 0 G77

,

B = (Bmn) =

0 B12 0 · · · 0

B21 0

0 0...

.... . .

0 · · · 0

=

0 Υ9 0 · · · 0

−Υ9 0

0 0...

.... . .

0 · · · 0

, (4.9)

where G44 = |α11

α4|. The solution presented in [5] is valid only at large distances, and

again it should be modified at finite distances. These modifications are given by (4.3),

(4.4) and (4.5) as well as by

g =D

H, Υ2 =

(n+ 1)α9

2πai(f − f) , Υ9 =

(n+ 1)α2

2πai(f − f) , (4.10)

where D =

√|α4α11|√|α2α9|

. It can again be checked that this modified background satisfies the

Killing spinor equations (3.2) for any arbitrary holomorphic f . The associated Killing

spinor is given by (3.6) and (3.8) subject to two additional constraints on χ given in [5].

Finally, consider the cases where n = 3 and n = 4. The associated Killing spinors both

have 1/8 of 16 real degrees of freedom. The solutions are characterised by the addition of

one (two) additional off-diagonal entries in the internal metric Gmn and in Bmn for n = 3

(n = 4) [5]. It is straightforward to modify these solutions along similar lines as the ones

discussed above, and again it can be checked that these modified solutions satisfy the

Killing spinor equations (3.2).

We thus see that, in any of the above modified solutions, the modifications are all encoded

in one single holomorphic function f .

Next, we would like to determine the holomorphic function f by demanding that the

modified solution have finite energy [4]. This will also render the solutions well behaved

at finite distances.

10

Let us first compute the energy carried by the modified solutions discussed above. The

integral (2.12) is computed to be (in units where 8πGN = 1)

E = i 2n∫dωdω

∂ωf∂ωf

(f + f)2= i 2n

∫dzdz

∂zf∂z¯f

(f +¯f)2

, (4.11)

where ω = a ln z and where we have introduced f = n+1πf for later convenience.

There is an elegant mechanism [9] for rendering the integral (4.11) finite. Let us take z

to be the coordinate of a complex plane. Then there is a one-to-one map from a certain

domain F on the f -plane (the so called ‘fundamental’ domain) to the z-plane. This map

is known as the j-function, j(f) = z. By means of this map, the integral (4.11) can

be pulled back from the z-plane to the domain F (the z-plane covers F exactly once).

Then, by using integration by parts, this integral can be related to a line integral over

the boundary of F, which is evaluated to be [9]

E = 2n2π

12= 2n

π

6(4.12)

and, hence, is finite.

By expanding j(f) = e2πf + 744 + O(e−2πf ) = z we see that f(ω) =1

2(n+1)(ω − 744e−ω + . . .), which indeed reproduces the correct asymptotic behaviour at

ω →∞. Thus we see that, by demanding the asymptotic behaviour of f to be modified

to f = πn+1

j−1(z), the associated energy becomes finite.

We note that the solutions discussed above represent one-center solutions. They can be

generalised to multi-center solutions via j(f(z)) = P (z)/Q(z), where P (z) and Q(z) are

polynomials in z with no common factors. These are the analogue of the multi-string

configurations discussed in [9].

It can be checked that the curvature scalar R ∝ ∂ωf∂ωf blows up at the special point

f = 1 (at this point, the j-function and its derivatives are given by j = 1728, j′ = 0),

whereas it is well behaved at the point f = eiπ/6 (at this point, the j-function and its

derivatives are given by j = j′ = j′′ = 0).

5 Supersymmetric solutions carrying one electric charge

Here we will first review the supersymmetric solutions constructed in [5] carrying one

electric charge. These solutions need again to be modified at finite distances.

The solutions in this class are also labelled by an integer n (n = 1, 2, 3, 4). Here the

11

dilaton was [5]

e2φ =n+ 2

2a ln r, (5.1)

whereas the associated space–time line element was

ds2 = −dt2 +a2

r2

(2a ln r

n+ 2

)n(dr2 + r2dθ2). (5.2)

Here, a = n+24π|αi|, where αi denotes the electric charge carried by the solutions. The

i-th component of the internal metric Gmn was given by Gii = 2a ln rn+2

. The gauge fields

strength, or equivalently Ψi+7, was taken to have the asymptotic form given in (2.16).

Introducing complex coordinates, ω = a ln z = a(ln r + iθ), then yields

ds2 = −dt2 + Hndωdω , e−2φ = H , Gii = H , (5.3)

where H = 2a ln rn+2

= ω+ωn+2

. The solution again becomes ill defined as r→ 1 and thus needs

to get modified at finite distances.

The asymptotic form of H suggests the replacement

H =ω + ω

n+ 2−→ H = f(ω) + f(ω) . (5.4)

Similarly, the asymptotic form of Ψi+7 given in (2.16) suggests the replacement

Ψi+7 =αi

4πai(ω − ω) −→ Ψi+7 =

(n+ 2)αi4πa

i(f − f) = ηαi i(f − f) , (5.5)

where ηαi denotes the sign of the charge αi. Denoting the imaginary part of f by Ψ,

Ψ = −i(f − f), yields

f =1

2

(e−2φ + iΨ

)(5.6)

as well as

Ψi+7 = −ηαiΨ . (5.7)

The associated field strength is given by

F(1)itβ = ηαi

∂βH

H2, β = r, θ , (5.8)

where ω = a(ln r + iθ) = r + iθ.

We will now discuss the solutions in more detail. Let us first consider the case where

n = 1. The associated Killing spinor has 1/2 of 16 real degrees of freedom. The solution

is characterised by the fact that the internal vielbein eam is diagonal and that Bmn = 0 [5].

12

The solution presented in [5], which has line element (5.2), should be modified to render

the solution well behaved at finite distances. This modification is given by (5.3), (5.4)

and (5.5). It can then be checked that the Killing spinor equations (3.2) are satisfied

for any arbitrary holomorphic f . The Killing spinor is given by (3.6) and (3.8) with

p = −1, subject to one additional condition on χ given in [5]. Thus, the requirement of

supersymmetry alone does not determine f .

Next, consider the case where n = 2. The associated Killing spinor has 1/4 of 16 real

degrees of freedom. The solution is characterised by the fact that now there is one non-

constant entry in the Bmn-matrix. The internal metric Gmn stays diagonal, however. As

an example, consider the case where the electric charge is taken to be α4, and where the

background fields Gmn and Bmn are given by [5]

G−1 =

G11 0 0 0 · · · 00 G22 0 0 · · · 00 0 G33 0 · · ·

0 0 0 G44 0...

.... . .

0 · · · 0 G77

=

g21 0 0 0 · · · 00 g2

2 0 0 · · · 00 0 G33 0 · · ·

0 0 0 1H

0...

.... . .

0 · · · 0 G77

,

B =

0 B12 0 · · · 0B21 0

0 0...

.... . .

0 · · · 0

=

0 Υ9 0 · · · 0−Υ9 0

0 0...

.... . .

0 · · · 0

. (5.9)

The solution presented in [5] is valid only at large distances, and it should be modified

at finite distances. These modifications are given by (5.3), (5.4) and (5.5) as well as by

g21 =

D21

H, g2

2 =D2

2

H, Υ9 =

(n+ 2)α2

4πai(f − f) , (5.10)

where D1D2 = 4πa(n+2)|α2|

. It can again be checked that this modified background satisfies

the Killing spinor equations (3.2) for any arbitrary holomorphic f . The associated Killing

spinor is given by (3.6) and (3.8) with p = −1, subject to two additional constraints on

χ given in [5].

Finally, consider the cases where n = 3 and n = 4. The associated Killing spinors both

have 1/8 of 16 real degrees of freedom. The solutions are characterised by the addition

of one (two) additional off-diagonal entries in Bmn for n = 3 (n = 4) [5]. In both cases,

the internal metric Gmn stays diagonal. It is straightforward to modify these solutions

along similar lines as the ones discussed above, and it can be checked that these modified

solutions satisfy the Killing spinor equations (3.2).

The modifications are again all encoded in one single holomorphic function f .

13

Next, we would like to determine the holomorphic function f by demanding that the

modified solution should have finite energy [4]. This time, computing the integral (2.12)

yields (in units where 8πGN = 1)

E = i n∫dωdω

∂ωf∂ωf

(f + f)2= i n

∫dzdz

∂z f∂z¯f

(f +¯f)2

, (5.11)

where ω = a ln z and where this time f = n+22πf .

As before, by using the j-function map j(f) = z, the energy (5.11) can be made finite

and is given by

E = nπ

6. (5.12)

Note that this is half the amount of energy carried by the solutions with two electric

charges that we discussed in section 4.

By expanding j(f) = e2πf + 744 + O(e−2πf ) = z we see that f(ω) =1

n+2(ω − 744e−ω + . . .), which indeed reproduces the correct asymptotic behaviour at

ω →∞. Thus we see that, by demanding the asymptotic behaviour of f to be modified

to f = 2πn+2

j−1(z), the associated energy becomes finite.

We note that the curvature scalar R blows up at the special point f = 1.

The solutions discussed above represent one-center solutions. They can again be gen-

eralised to multi-center solutions via j(f(z)) = P (z)/Q(z), where P (z) and Q(z) are

polynomials in z with no common factors.

6 Eleven dimensional interpretation

Heterotic string theory compactified on a seven-torus is related to M-theory compactified

on S1/Z2×T7 [18]. Hence, our solutions should have an eleven dimensional interpretation

in terms of configurations of intersecting membranes (M2), 5-branes (M5), M-waves and

Kaluza–Klein M-monopoles, which are all supersymmetric solutions to the low-energy

effective action of M-theory. These four basic solutions all preserve 1/2 of the eleven-

dimensional supersymmetry. (For a review see [3, 19, 20] and references therein.) In ten-

dimensional heterotic string theory, the basic supersymmetric solutions, when reducing

from eleven dimensions, are a fundamental string (coming from M2), a wave, a KK

monopole and a NS 5-brane. Each of these ten-dimensional basic solutions preserve

1/2 of 16 supersymmetry. Some of the configurations that one gets when considering

various combinations of these objects, when reduced down to three dimensions, should

14

correspond to the solutions constructed in [5]. The following interpretation emerges

when comparing the ten-dimensional metric of each of these four basic building blocks

with the ten-dimensional ansatz for the vielbein (2.4): the wave, when reduced to three

dimensions, gives rise to solutions with non-zero A(1)mµ , which are the gauge fields coming

from the reduction of G(10)MN ; the fundamental string gives rise to solutions with non-

zero A(2)µm, coming from the reduction of B

(10)MN ; the NS 5-branes give rise to solutions

with non-zero off-diagonal internal Bmn components and the Kaluza–Klein monopoles to

solutions with non-zero off-diagonal internal metric Gmn. We will now review the four

basic supersymmetric solutions and we will discuss their reduction to three dimensions

in detail.

The eleven-dimensional M-branes, M-waves and M-monopoles are solutions to the low-

energy effective action of M-theory, given by D = 11 supergravity. The bosonic action

contains a metric gMN and a three-form potential AMNP , with field strength FMNPQ =

24∇[MANPQ] :

S =∫ √−gR−

1

12F 2 −

1

432εM1...M11FM1...M4FM5...M8AM9...M11. (6.1)

Supersymmetric solutions to the equations of motion of this action can be obtained by

looking for backgrounds that admit 32-component Majorana spinors ε for which the

supersymmetry variation of the gravitino field ψM vanishes.

The M2-brane solution has the form [21]

ds211 = H1/3[

1

H(−dt2 + dx2

1 + dx211) + (dx2

2 + . . . dx29)] (6.2)

with

Ft 1 11 α =c

2

∂αH

H2, H = H(x2, . . . , x9), ∇2H = 0, c = ±1. (6.3)

The solution admits Killing spinors ε = H−1/6 η with the constant spinor η satisfying

Γ0 1 11 η = c η, where Γ0...p ≡ Γ0 . . . Γp is the product of p+ 1 distinct Gamma matrices in

an orthonormal frame. Given that (Γ0 1 11)2 = 1 and Tr Γ0 1 11 = 0, this solution admits

16-component Killing spinors and preserves half of the supersymmetry.

The single harmonic function determining the solution depends on the orthogonal direc-

tions to the 2-brane, ~x = x2, . . . , x9. The M2-brane carries electric four-form charge Qe

defined as the integral of the seven-form ∗F around a seven-sphere that surrounds the

brane. c = 1 corresponds to a M2-brane and c = −1 to an anti-M2-brane.

We now dimensionally reduce the membrane to ten dimensions to obtain the fundamental

string (NS1) by using that [22]

ds211 = e2/3Φ(10)

dx211 + e−1/3Φ(10)

ds210. (6.4)

15

Thus, e2Φ(10)= H−2, and

ds210 = H−1(−dt2 + dx2

1) + dx22 + . . .+ dx2

9. (6.5)

Here, ds210 denotes the ten-dimensional line element in the string frame. In order to

dimensionally reduce the fundamental string to three dimensions, we take H to be a

function of two transverse coordinates only, H = H(x8, x9), subject to ∇2H = 0, where

∇2 now denotes a two-dimensional flat Laplacian. We can take the coordinates x8 and x9

to be either cartesian or cylindrical. Both coordinate choices, being related by a conformal

transformation, are compatible with ∇2 being a flat Laplacian. In the following, we take

x8 and x9 to be cylindrical coordinates (ω = x8 + ix9 = r + iθ).

The dimensional reduction of the fundamental string solution to three-dimensions is now

obtained by demanding that the line element take the form

ds210 = e2φ(3)

gE(3)µν dxµdxν +G(7)

mndymdyn, (6.6)

where G(7)mn is the seven-dimensional internal metric with internal coordinates ym, and

gE(3)µν is the three-dimensional Einstein metric. Compatibility of the ten-dimensional ac-

tion and of the three-dimensional action in the Einstein frame requires that e2φ(3)=

e2Φ(10)(detG(7)

mn)−1 = H−1 and the three-dimensional space–time line element in the Ein-

stein frame is

ds2 = −dt2 +Hdωdω. (6.7)

In D = 10, the fundamental string couples to the antisymmetric 2-tensor with Btx1 ∝

H−1, which in D = 3 yields that A(2)tm = c

2H−1, or F

(2)tαm = c

2∂αHH2 , where m = 1.

The solution corresponding to the eleven-dimensional M5-brane is of the form [23]

ds211 = H2/3[

1

H(−dt2 + dx2

1 + . . . dx25) + (dx2

6 + . . .+ dx29 + dx2

11)] (6.8)

with

Fα1...α4 =c

2εα1...α5∂α5H, H = H(x6, . . . , x9, x11), ∇2H = 0, c = ±1, (6.9)

where εα1...α5 is the flat D = 5 alternating symbol.

The solution admits 16-component Killing spinors ε = H−1/2η with η satisfying

Γ012345 η = cη. The M5-brane carries magnetic four-form charge Qm obtained by in-

tegrating F around a four-sphere that surrounds the M5-brane. Here again, c = ±1

corresponding to a M5- and an anti-M5-brane respectively.

Using (6.4), we find that the M5-brane reduces to a NS 5-brane in ten dimensions, with

metric

ds210 = −dt2 + dx2

1 + . . .+ dx25 +H(dx2

6 + . . .+ dx29). (6.10)

16

Here eΦ(10)= H. Setting ω = x8 + ix9 and taking H = f(ω) + f(ω), we find that, by

using (6.6), the three-dimensional line element is again given by (6.7), with eφ(3)

= 1.

Reducing Fα1α2α3α4 down to three dimensions gives rise to H67α = ∂αB67 with H67r ∝

∂θH = i∂r(f − f), H67θ ∝ −∂rH = i∂θ(f − f). So it follows that B67 ∝ i(f − f) which

is in accordance with the expressions for the internal Bmn field given in sections 4 and 5.

The wave solution in ten dimensions is given by the metric (with eφ(10)

= 1) [24]

ds210 = −dt2 + dy2

1 + (H − 1)(dt− dy1)2 + (dx2

2 + . . .+ dx29), (6.11)

or, with y1 = t+ cx1 ,

ds210 = 2cdtdx1 +Hdx2

1 + (dx22 + . . .+ dx2

9). (6.12)

This corresponds precisely to our n = 1 solutions carrying one electric charge, with a

gauge field A(1)aµ = A(1)m

µ eam coming from the reduction of the ten-dimensional metric

G(10)MN . Indeed, the ten-dimensional line element (2.4) used in the reduction is

ds210 = (e2φ(3)

gE(3)µν +A(1)a

µ ηabA(1)bν )dxµdxν + 2A(1)a

µ ηabebndx

µdyn +G(7)mndy

mdyn. (6.13)

Inserting (5.3) and (5.8) into (6.13), we recover (6.12), with c = ηα1 , where x1, x2 . . . x7

belong to the seven-dimensional torus T7 and ω = x8 + ix9.

The Kaluza–Klein monopole in ten dimensions is given by the metric [25]

ds210 = −dt2 +Hdx2

i +H−1(dz +Aidxi)2 + dx2

1 + . . .+ dx25, i = 6, 8, 9, (6.14)

with H = H(xi), Fij = ∂iAj − ∂jAi = c εijk∂kH, e2Φ(10)

= 1, c = ±.

In five dimensions, the metric reduces to

ds25 = −dt2 +Hdx2

i +H−1(dz +Aidxi)2, (6.15)

with e2φ(5)= 1.

Now, let H = H(x8, x9). We can set A8 = A9 = 0. Then, ∂8A6 = −c∂9H, ∂9A6 = c∂8H.

The metric is now

ds25 = −dt2 +H(dx2

8 + dx29) +Hdx2

6 +H−1(dz +A6dx6)2

= e2φ(3)

gE(3)µν dxµdxν +Gmndy

mdyn, (6.16)

with eφ(3)

= 1 and the off-diagonal internal metric is given by

Gmn =

H +A26H−1 A6H

−1

A6H−1 H−1

. (6.17)

We now present the supersymmetric heterotic solutions discussed in sections 4 and 5 as

dimensionally reduced solutions corresponding to orthogonally intersecting strings, NS

5-branes, waves and KK monopoles in ten dimensions.

17

6.1 Solutions carrying two electric charges

As discussed in section 4, the three-dimensional space–time line element in the Einstein

frame takes the form

ds2 = −dt2 +H2ndωdω. (6.18)

Consider first the case n = 1. This solution with diagonal and constant internal vielbein

eam and Bmn = 0 corresponds to a fundamental string and a wave in D = 10, with the

wave travelling along the string.

In D = 10, the solution is

ds210 = (Hs)

−1[−dt2 + dy21 + (Hw − 1)(dt− dy1)2] + dx2

2 + . . .+ dx29 (6.19)

or, using (6.12)

ds210 = (Hs)

−1[ 2cdtdx1 +Hwdx21] + dx2

2 + . . .+ dx29. (6.20)

The ten dimensional dilaton is given by eΦ(10)= H−1

s . Note that the line element reduces

to the wave (6.12) or string (6.5) line elements when we respectively set Hs and Hw to

1. We now set Hs = Hw = H(x8, x9). Comparing (6.20) with (6.13), we find that the

internal metric is constant and that consequently eφ(3)

= H−1. Furthermore the gauge

field is given by A(1)at = A

(1)mt eam = cH−1, where a = 1. Note that this is in accordance

with (4.3) and (4.8).

Since the wave travels along the string, they are both characterized by Killing spinors

which obey the same condition Γ01 η = c1 η. This solution thus preserves 1/2 of 16

supersymmetry.

Note however that, as described in [5], this purely electric solution can be dualized to a

solitonic solution represented by one off-diagonal entry in the internal metric as well as

one non-constant entry in the internal Bmn-matrix. These off-diagonal contributions play

the role of ‘magnetic’ charge from the ten-dimensional point of view and correspond re-

spectively to adding a KK monopole and a NS 5-brane, with a common 5+1-dimensional

worldvolume. (This is one of two possible intersection patterns of a KK monopole and

a NS 5-brane in D = 10 [20].) Taking the 5-brane to be oriented along the hyperplane

3, 4, 5, 6, 7, this configuration is characterized by Γ034567 η = c2η and Γ1289 η = c3η. The

5-brane and monopole each break 1/2 of 16 supersymmetry. Since, however, the prod-

uct of the two Gamma matrix projection operators gives the ten-dimensional chirality

operator Γ11, this pair preserves again 1/2 of 16 supersymmetry. (The ten-dimensional

chirality operator, in our notation [5], is Γ11 = ±γ0γ1γ2γ4⊗ iI8. Γ11ε = ε is then equiva-

lent to conditions (3.7)). The ten-dimensional line element of this configuration is given

18

by

ds210 = −dt2 +dx2

3 + . . .+dx27 +H5H

−1KK(βdz+A2dx2)2 +H5HKK(dx2

2 +dx28 +dx2

9). (6.21)

Note that this line element reduces to (6.10) and (6.14) (up to a relabelling of the co-

ordinates) when we respectively set HKK and H5 to 1. The ten-dimensional dilaton is

given by eΦ(10)= H5. Setting H5 = HKK = H(x8, x9), and comparing with (6.6), yields

a constant three-dimensional dilaton as well as the three-dimensional line element (6.18)

with n = 1. Furthermore, the non-constant part of the internal metric Gmn is found here

to be

Gmn =

β2 βA2

βA2 A22 +H2

, (6.22)

where β =√G11 and where ∂8A2 = −c∂9H, ∂9A2 = c∂8H. For H = f + f , it then follows

that A2 = −ic (f − f). This is indeed the off-diagonal internal metric given in [5].

Consider next the solution with n = 2. There is now, in addition to the two electric

charges (corresponding to having a wave and a string), one non-constant off-diagonal

entry in the internal metric as well as one non-constant entry in the Bmn-matrix. In view

of the discussion above, this amounts to adding a NS 5-brane and a KK monopole to

(6.20). The whole configuration preserves 1/4 of 16 supersymmetry. The ten-dimensional

line element thus includes a wave, a fundamental string, a 5-brane and a monopole. The

wave is along the string and both are in the worldvolume of the NS 5-brane and KK

monopole. The 5-brane and monopole have common worldvolumes. The associated line

element is given by

ds210 = H−1

s [2cdtdx1 +Hwdx21] + dx2

2 + . . .+ dx25 +H5H

−1KK(dx6 +A7dx7)2

+ H5HKK(dx27 + dx2

8 + dx29) , (6.23)

and the ten-dimensional dilaton is given by eΦ(10)= H−1

s H5. We now identify Hs = Hw =

H5 = HKK = H(x8, x9). It is easy now to check that the ten-dimensional line element

gives rise to the correct three-dimensional space–time metric (6.18) with n = 2 as well

as to the off-diagonal internal metric given in (4.9). This is done by sending x7 −→x7

D

(where D =√|α4α11|√|α2α9|

) and by relabelling the coordinates.

Consider next the solution with n = 3. This solution is obtained from the n = 2 solution

by adding an additional off-diagonal entry in both the internal metric Gmn and in the

Bmn-matrix. This amounts to adding an additional NS 5-brane and a KK monopole to

the line element (6.23) in the following way:

ds210 = H−1

s [2cdtdx1 +Hwdx21] + dx2

2 + dx23 +H5H

−1KK(dx4 +A5dx5)2 +H5HKKdx

25

+ H5H−1KK(dx6 +A7dx7)2 +H5HKKdx

27 +H2

5H2KK(dx2

8 + dx29) . (6.24)

19

Note that the 5-branes intersect orthogonally in a 3-brane [19] and that the KK monopoles

have a common three-dimensional worldvolume [20]. The ten-dimensional dilaton is given

by eΦ(10)= H−1

s H25 . This configuration preserves 1/8 of 16 supersymmetry.

Finally consider our solution with n = 4. It is obtained by adding an additional off-

diagonal entry both in the Gmn sector and in the Bmn sector. This is achieved by adding

yet an additional monopole and an additional 5-brane. The resulting ten-dimensional

line element is given by

ds210 = H−1

s [2cdtdx1 +Hwdx21] +H5H

−1KK(dx2 +A3dx3)2 +H5HKKdx

23

+ H5H−1KK(dx4 +A5dx5)2 +H5HKKdx

25 +H5H

−1KK(dx6 +A7dx7)2

+ H5HKKdx27 +H3

5H3KK(dx2

8 + dx29) . (6.25)

The ten-dimensional dilaton is given by eΦ(10)= H−1

s H35 . Note that adding these two

additional objects does not break any further supersymmetry, since the Gamma projec-

tion operator for the last 5-brane is given by the product of the Gamma operators of

the wave and the two other 5-branes. Thus, this configuration also preserves 1/8 of 16

supersymmetry.

6.2 Solutions carrying one electric charge

In this case, the space–time line element in the Einstein frame has the form

ds2 = −dt2 +Hndωdω. (6.26)

The n = 1 solution corresponds in ten dimensions to a wave and its reduction is the same

as in (6.13) and below. The wave preserves 1/2 of 16 supersymmetry.

The n = 2 case can be described in terms of a wave and one NS 5-brane, since a non-

constant entry in the Bmn matrix is added. The ten-dimensional metric describing this

pair is given by

ds210 = 2cdtdx1 +Hwdx

21 + dx2

2 + dx23 + dx2

4 + dx25 +H5(dx2

6 + . . .+ dx29) (6.27)

and the ten-dimensional dilaton by eΦ(10)= H5.

This solution preserves 1/4 of 16 supersymmetry. Reducing according to (6.13), we

recover our solution given in (5.9) (up to relabelling of the coordinates) with Hw = H5 =

H(x8, x9), e−2φ(3)= H, A

(1)a=1t = c/

√H, and G11 = G66 = G77 = H.

The n = 3 (n = 4) case corresponds in the same way to two (three) orthogonally

intersecting 5-branes and a wave, where each pair of 5-branes intersects over a 3-brane

20

[19]. For instance, the ten-dimensional line element for the n = 4 case is given by

ds210 = 2cdtdx1 +Hwdx

21 +H5(dx2

2 +dx23)+H5(dx2

4 +dx25)+H5(dx2

6 +dx27)+H3

5 (dx28 +dx2

9)

(6.28)

and the ten-dimensional dilaton by eΦ(10)= H3, where we have again identified the

harmonic functions Hw = H5 = H(x8, x9).

By using the same argument as in the previous subsection, we conclude that both these

cases preserve 1/8 of 16 supersymmetry.

7 Conclusions

We showed that the static supersymmetric solutions constructed in [5] can be turned

into finite energy solutions, which we computed. The energy of the solutions carrying

one electric charge was found to be given by E = n π6, whereas the energy of the solutions

carrying two electric charges was found to be E = 2n π6.

The U-duality group of the low-energy heterotic theory is O(8, 24;Z) [4]. Solutions which

are obtained by O(8, 24;Z) transformations from the ones discussed above will also have

finite energy. For instance, the compactified heterotic string solutions of [7, 8] will have

the same energies as the solutions carrying one electric charge discussed in section 5,

and similarly for the three-dimensional solutions obtained by compactifying a wave and

up to three NS 5-branes and Kaluza–Klein monopoles intersecting orthogonally in ten

dimensions.

The curvature scalar R associated with the solutions discussed in sections 4 and 5 is

regular everywhere with the exception of the special point φ = 0. It would be interesting

to understand the physics at this special point in moduli space further.

Acknowledgement

We would like to thank I. Bakas, K. Behrndt, R. Khuri, D. Lust, S. Mahapatra, T.

Mohaupt and J. Schwarz for valuable discussions. One of us (G. L. C.) is grateful for the

hospitality at the Institute of Physics of the Humboldt University Berlin, where part of

the work was carried out.

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21

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[4] A. Sen, Nucl. Phys. B434 (1995) 179, hep-th/9408083.

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[6] N. Marcus and J. Schwarz, Nucl. Phys. B228 (1983) 145.

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